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A previous post in this series introduced us to two-dimensional monad theory, where we were told about 22-monads, their strict algebras, and the interplay of the various morphisms that can be considered between them. The paper of Lack has a slightly different focus in that not only are we interested in morphisms of varying levels of strictness but also in the weaker notions of algebra for a 22-monad, namely the pseudoalgebras and lax algebras.

An example that we will consider is that of the free monoid 22-monad on the 22-category Cat\mathbf{Cat} of small categories, functors, and natural transformations. The strict algebras for this 22-monad are strict monoidal categories, whilst the lax algebras are (unbiased) lax monoidal categories. Similarly, the pseudoalgebras are (unbiased) monoidal categories. The classic coherence theorem of Mac Lane is then almost an instance of saying that the pseudoalgebras for the free monoid 22-monad are equivalent to the strict algebras. We will see conditions for when this can be true for an arbitrary 22-monad.

Thanks go to Emily, my supervisor Nick Gurski, the other participants of the Kan extension seminar, as well as all of the participants of the Sheffield category theory seminar.

Example We’ll see what’s going on by looking at the free monoid 22-monad again, call it MM. A lax algebra for MM is a category XX and a functor x:MX→Xx : MX \rightarrow X with natural transformations χ\chi, χ0\chi_0 as above. Now MXMX is the coproduct
∐n∈ℕXn
\coprod_{n \in \mathbb{N}} X^n
meaning that objects in MXMX are finite lists of objects in XX, and similarly for morphisms. The functor x:MX→Xx : MX \rightarrow X is a functor out of a coproduct so in fact corresponds to a family of functors
(xn:Xn→X)n∈ℕ
(x_n : X^n \rightarrow X)_{n \in \mathbb{N}}
which we can view as being the nn-ary tensors of an unbiased lax monoidal category. The natural transformation χ\chi then has components which are morphisms
((a11⊗…⊗a1k1)⊗…⊗(an1⊗…⊗ankn))→(a11⊗…⊗ankn)
\left(\left(a_{11} \otimes \ldots \otimes a_{1k_1}\right) \otimes \ldots \otimes \left(a_{n1} \otimes \ldots \otimes a_{nk_n}\right)\right) \rightarrow \left(a_{11} \otimes \ldots \otimes a_{nk_n}\right)
in XX. These are what correspond to the associators in a biased monoidal category. The associativity and unit axioms can then be found to be expressed by the lax algebra axioms.

These differing levels of strictness offer us a whole host of 22-categories to look at. For our purposes we will be looking at the following 22-categories:

The second section of the paper begins by considering lax morphisms of the form
(f,f¯):(X,x,χ,χ0)→(Y,y),
(f, \overline{f}) : (X, x, \chi, \chi_0) \rightarrow (Y,y),
between a lax algebra XX and a strict algebra YY. The idea is that lax morphisms of this form in Ps-T-Alg\text{Ps-}T\text{-Alg} can be recast as strict morphisms
(g=y⋅Tf,g¯=1y*Tf¯):(TX,μX)→(Y,y)
(g = y \cdot Tf, \overline{g} = 1_{y} \ast T\overline{f}) : (TX, \mu_X) \rightarrow (Y,y)
in T-AlgsT\text{-Alg}_s. There is an inclusion 2-functor
U:T-Algs→Lax-T-Algl
U : T\text{-Alg}_s \rightarrow \text{Lax-}T\text{-Alg}_l
and the aim is to construct a left adjoint. To this end, Lack describes a universal property related to 11-cells in T-AlgsT\text{-Alg}_s of the form TX→X′TX \rightarrow X' so that there is an isomorphism
T-Algs(X′,Y)≅Lax-T-Algl(X,Y)
T\text{-Alg}_s(X',Y) \cong \text{Lax-}T\text{-Alg}_l(X,Y)
which is natural in Y. This tells us that if such an object X′X' exists for every lax algebra XX, then the left adjoint also exists.

Consider for a moment, an algebra (A,a)(A,a) for a 11-monad SS on a 11-category 𝒞\mathcal{C}. We know that this can be expressed as the reflective coequaliser of the diagram
⟶μAS2A⟵SηASA⟶Sa
\begin{array}{ccc}
\quad & \overset{\mu_A}{\longrightarrow} & \quad \\
S^2A & \overset{S\eta_A}{\longleftarrow} & SA \\
\quad & \overset{Sa}{\longrightarrow} & \quad \\
\end{array}
in the category S-AlgS\text{-Alg} of SS-algebras. However in the case of a lax algebra (X,x,χ,χ0)(X, x, \chi, \chi_0) for a 22-monad TT, this won’t be the case. Instead we can form lax coherence data
→μTA→μAT3X→TμXT2X←TηATX→T2x→Tx
\begin{array}{ccccc} \quad & \overset{\mu_{TA}}{\rightarrow} & \quad & \overset{\mu_A}{\rightarrow} & \\ T^3X & \overset{T\mu_X}{\rightarrow} & T^2X & \overset{T\eta_A}{\leftarrow} & TX\\ \quad & \overset{T^2x}{\rightarrow} & \quad & \overset{Tx}{\rightarrow} & \end{array}
in T-AlgsT\text{-Alg}_s when we accompany it with 22-cells Tχ0T\chi_0 and TχT\chi, where the rest of the 22-cells are just identities arising from the 22-monad axioms. The universal property alluded to above is then that the lax codescent object of this lax coherence data is the same as that of the replacement (strict) algebra X′X' which would give the adjunction previously described.

If all of the mentions of 22-cells in the above description of a lax codescent object were replaced with invertible22-cells, then we would have the notion of a codescent object. This is the analogous situation in the case of pseudoalgebras, where the aim is to find a left adjoint to the inclusion to the inclusion 22-functor
T-Algs→Ps-T-Alg.
T\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg}.

A useful observation is that lax codescent objects may be defined using weighted colimits and can be built from coinserters and coequifiers. Also worthy of note is that codescent objects can be built from co-iso-inserters and coequifiers. Now co-iso-inserters exist whenever coinserters and coequifiers do, so that anything we want to prove about lax algebras by utilising such colimits, will also be true for pseudoalgebras.

This section of the paper also includes a number of results concerning adjunctions between the various 22-categories of algebras, with the following theorem then being the basis for the first characterisation of a coherence theorem.

Theorem: (Lack, 2.4) For a 22-monad TT on a 22-category 𝒦\mathcal{K}, the inclusion T-Algs→Lax-T-AlglT\text{-Alg}_s \rightarrow \text{Lax-}T\text{-Alg}_l has a left adjoint if any of the following conditions holds:

𝒦\mathcal{K} is cocomplete and TT preserves α\alpha-filtered colimits for some regular cardinal α\alpha.

Conditions 22 and 33 also give us a left adjoint to the inclusion T-Algs→Ps-T-AlgT\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg}. Furthermore, we also find that a left adjoint to the inclusion T-Algs→T-AlgT\text{-Alg}_s \rightarrow T\text{-Alg}, which we saw in the paper of Blackwell, Kelly, and Power, also exists under these conditions. Something else that we saw in that paper is the reason for needing TT to preserve these colimits - the colimits exist in T-AlgsT\text{-Alg}_s just when TT preserves them.

Coherence

The simplest possible characterisation of coherence for 22-monads would be:

Theorem-Schema: The inclusion T-Algs→Ps-T-AlgT\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg} has a left adjoint, and the components of the unit are equivalences in Ps-T-Alg\text{Ps-}T\text{-Alg}.

Something that is rather nice, though, is that we already have some conditions under which the theorem-schema is satisfied.

Theorem: (Lack, 3.2) If TT is a 22-monad on a 22-category 𝒦\mathcal{K} admitting codescent objects, and TT preserves them, then the inclusion T-Algs→Ps-T-AlgT\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg} has a left adjoint, and the components of the unit are equivalences in Ps-T-Alg\text{Ps-}T\text{-Alg}. In particular this is the case if 𝒦\mathcal{K} has coinserters and coequifiers, and TT preserves them.

The proof of this is rather simple and falls out of the two-dimensional universal property of the codescent objects.

I’m going to roll the latter two sections of the paper together now and talk about the other characterisation of coherence, which concerns a general coherence result of Power. That paper looks at 22-monads on CatX\mathbf{Cat}^X and CatgX\mathbf{Cat}^X_g, where XX is a small set and the latter 22-category is attained from the first by only considering invertible 22-cells. Power then shows that if TT is a 22-monad on one of these 22-categories which preserves bijective-on-objects functors, then every pseudoalgebra for TT is equivalent to a strict one.

Some 22-monads which satisfy these conditions include Set\mathbf{Set}-based clubs, whose strict algebras give such structures as monoidal categories (see the scope of the results below for more monoidal examples) or categories with strictly associative finite products or coproducts. Also described in Power’s paper is a 22-monad on CatX×X\mathbf{Cat}^{X \times X} for which the pseudoalgebras are unbiased bicategories with object set XX. The coherence result then tells us that every bicategory is biequivalent to a 22-category with the same set of objects.

Comparing Power’s statement to the theorem-schema, we see that they are not quite the same. The schema asks for there to be an adjunction for which the components of the unit give the equivalences we are concerned with. As it turns out, the conditions which Power proposes are indeed enough to give what we desire, and this is what the latter characterisation of Lack looks at.

Recall that every functor can be factored as a bijective-on-objects functor followed by a full and faithful functor. This gives an orthogonal factorisation system(bo,ff)(bo,ff) on Cat\mathbf{Cat}. However, the (bo,ff)(bo,ff) factorisation system has an extra two-dimensional property concerning 22-cells. If we are given a natural isomorphism
A⟶RCF↓⇓α↓GB⟶SD
\begin{matrix}
A & \overset{R}{\longrightarrow} & C \\
{}_{F}{\downarrow} & {\Downarrow}_{\alpha} & \downarrow^G \\
B & \underset{S}{\longrightarrow} & D \\
\end{matrix}
where FF is bijective-on-objects and GG is full and faithful, then there is a unique pair (H,β)(H,\beta) consisting of a functor H:B→CH:B \rightarrow C and a natural isomorphism β:GH⇒S\beta:GH \Rightarrow S such that HF=RHF = R and the whiskering of β\beta with FF gives back α\alpha. For an arbitrary 22-category 𝒦\mathcal{K}, an orthogonal factorisation system with such a property is deemed an enhanced factorisation system.

Theorem: (Lack, 4.10) If 𝒦\mathcal{K} is a 22-category with an enhanced factorisation system (ℒ,ℛ)(\mathcal{L},\mathcal{R}) having the property that if j∈ℛj \in \mathcal{R} and jk≅1jk \cong 1 then kj≅1kj \cong 1, and if TT is a 22-monad on 𝒦\mathcal{K} for which TT preserves ℒ\mathcal{L}-maps, then the inclusion T-Algs→Ps-T-AlgT\text{-Alg}_s \rightarrow \text{Ps-}T\text{-Alg} has a left adjoint, and the components of the unit of the adjunction are equivalences in Ps-T-Alg\text{Ps-}T\text{-Alg}.

It is interesting to see the scope of these results and the places in which people have considered this type of coherence problem before.

Dunn proved the theorem-schema when 𝒦\mathcal{K} is the 22-category of based topological categories and for which TT is a 22-monad induced by a braided Cat\mathbf{Cat}-operad.

The theorem-schema was also proved by Hermida, though required much more of both the 22-category 𝒦\mathcal{K} and the 22-monad TT, such as requiring existence and preservation of various limits and colimits, exactness properties relating these, as well as further conditions on the unit and multiplication of the 22-monad. Something that does fall out of this alternative setup is that TT can be replaced by a new 22-monad, on a different 22-category, which is lax-idempotent.

Rather more recently Nick Gurski and I wrote about operads with general groups of equivariance. Therein we showed that the 22-monads which arise from Cat\mathbf{Cat}-operads in this way satisfy the coherence conditions following the enhanced factorisation system route. These 22-monads capture many different structures, including monoidal categories, braided monoidal categories, symmetric monoidal categories, and ribbon braided monoidal categories. Thus we can say, for example, that every unbiased braided monoidal category is equivalent to a braided strict monoidal category, and similarly for the other variations.

The first theorem we mentioned above has three conditions, the third being the requirement that 𝒦\mathcal{K} is cocomplete and TT preserves α\alpha-filtered colimits for some regular cardinal α\alpha. We mentioned aboe that it was proved by Blackwell, Kelly, and Power that this is also sufficient to give a left adjoint to the inclusion U:T-Algs→T-AlgU : T\text{-Alg}_s \rightarrow T\text{-Alg}. They also proved further that if 𝒦\mathcal{K} is locally α\alpha-presentable then there is a 22-monad T′T' which preserves α\alpha-filtered colimits and where T′-Algs=Ps-T-AlgT'\text{-Alg}_s = \text{Ps-}T\text{-Alg}. The result of the theorem we discussed then follows when 𝒦\mathcal{K} is locally presentable and TT preserves α\alpha-filtered colimits. Lack comments that it is a major unsolved problem as to whether the entire theorem-schema can be shown to be true under these asumptions - and further whether it is true when 𝒦\mathcal{K} is only cocomplete.

Posted at June 2, 2014 12:46 AM UTC

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Re: Codescent Objects and Coherence

Another interesting appearance of codescent objects, as touched on in this paper, is in the bo-ff factorisation of a functor. Just as how in Set (or more generally a regular category) we may form the image factorisation of a function by taking the coequaliser of its kernel pair (the equivalence relation it induces on its domain), in Cat (or more generally) we get the bo-ff factorisation of a functor by taking the codescent object of its “higher kernel”. See for instance John Bourke’s thesis, where this 2-dimensional exactness play continues, featuring cateads in the role of equivalence relations.

Re: Codescent Objects and Coherence

Nick pointed something out to me this morning regarding the last bullet point above - some discussion related to this also came up between the seminar participants as well. Turns out I should have looked more closely at Mike Shulman’s paper showing that not every pseudoalgebra is equivalent to a strict one and not just looked at what the counterexample was. In fact, in the abstract, he says that the 22-monad giving the counterexample is finitary, i.e., preserves α\alpha-filtered colimits for some regular cardinal α\alpha. This is the the third condition of Theorem 2.4 in the post. In the last bullet point of the post I repeated what Steve had said in that it would be ideal to be able to prove the coherence theorem under the condition that 𝒦\mathcal{K} be cocomplete and TT preserve α\alpha-filtered colimits but Mike’s counterexample clearly shows this wouldn’t be sufficient.

Re: Codescent Objects and Coherence

I thought I’d create a thread similar the one we had before, in order to collect some more examples of where the coherence theorems apply.

A few neat examples come from some stuff I was looking at to do with TT-multicategories. I’ll try to describe them here.

A TT-multicategory is a multicategory defined relative to a cartesian monadTT, from which we recover plain multicategories as those defined relative to the free monoid monad on Set\mathbf{Set}. This can also be done for 22-cartesian 22-monads - replace every mention of pullback by 22-pullback. For now I just want to run with the identity 22-monad II on Cat\mathbf{Cat}, so we can see what’s going on.

To define an II-multicategory we need a category of objects, C0C_0, and a category of arrows, C1C_1, such that (along with some maps) these form a monad in the bicategory of spans, Span(Cat)Span(\mathbf{Cat}), in Cat\mathbf{Cat}. For our example take these to be the discrete categories C0={s,t}C_0 = \lbrace s, t\rbrace and C1=C0×C0C_1 = C_0 \times C_0. Once we have this data we can define a monad IcI_c on Cat/C0\mathbf{Cat}/C_0. Note that an object (X,rX:X→C0(X, r_X : X \rightarrow C_0 can be seen as the disjoint union of two categories, XsX_s and XtX_t, each lying over the respective object. Now we define Ic(X,rX)I_c(X, r_X) as being the composite running along the top of the following diagram, defined by using a 22-pullback.
IcX⟶C1⟶cC0↓↓dX⟶rXC0
\begin{array}{ccccc}
I_cX & \longrightarrow & C_1 & \overset{c}{\longrightarrow} & C_0 \\
\downarrow & \quad & \downarrow_d & & \\
X & \underset{r_X}{\longrightarrow} & C_0 & & \\
\end{array}
So the category IcXI_cX consists of triples (a,i1,i2)(a, i_1, i_2) where a∈Xa \in X and i1,i2∈C0i_1, i_2 \in C_0.

Now a strict algebra x:Ic(X,rX)→(X,rX)x : I_c(X,r_X) \rightarrow (X,r_X) is a functor (which I’ll also call xx) x:IcX→Xx : I_cX \rightarrow X, making the usual diagrams commute. It is also required to be a 11-cell in Cat/C0\mathbf{Cat}/C_0 which tells us that, for example, the object (a,s,t)(a, s, t) will end up in the category XtX_t. Similarly for the other possibilities (ss)(ss), (ts)(ts), and (tt)(tt). What this actually does is define four functors xs:Xs→Xsx_s : X_s \rightarrow X_s, xt:Xt→Xtx_t : X_t \rightarrow X_t, xst:Xs→Xtx_{st} : X_s \rightarrow X_t, and xts:Xt→Xsx_{ts} : X_t \rightarrow X_s, where the axioms tell us that xsx_s and xtx_t are both identities. We also see that xstxts=1Xtx_{st} x_{ts} = 1_{X_t} and xtsxst=1Xsx_{ts} x_{st} = 1_{X_s}, so that a strict IcI_c-algebra is in fact an isomorphism of categories. It is then easy to see that a pseudoalgebra for IcI_c is an equivalence of categories.

Now Cat/C0\mathbf{Cat}/C_0 inherits the (bo,ff)(bo,ff) enhanced factorisation system and IcI_c preserves bijective-on-objects functors, so IcI_c satisfies the coherence theorem. Thus every equivalence of categories is ‘equivalent’ to an isomorphism of categories. To spell this out in more detail we actually have a commuting square
Xs⟶≅Xt≃↓↓≃Xs′⟶≃Xt′
\begin{array}{ccc}
X_s & \overset{\cong}{\longrightarrow} & X_t \\
{\simeq}_{\downarrow} & \quad & \downarrow^{\simeq} \\
X_s' & \underset{\simeq}{\longrightarrow} & X_t' \\
\end{array}
where the bottom horizontal and both vertical functors are equivalences, and the top horizontal arrow is an isomorphism.

Now we can do a similar thing to this for monoidal functors, which we acquire by using the free monoid 22-monad MM on Cat\mathbf{Cat}. However, instead of setting C1=C0×C0C_1 = C_0 \times C_0 we now have it be a subcategory of MC0×C0MC_0 \times C_0 generated by tuples of the form (s,…,s;s)(s, \ldots, s ; s), (t,…,t;t)(t, \ldots, t; t) and (s,…,s;t)(s, \ldots, s ; t), as well as those with empty ‘domain’. After a bit of a slog it’s possible to see that strict algebras for the induced 22-monad McM_c are in fact strict monoidal functors (just one functor Xs→XtX_s \rightarrow X_t this time, rather than getting one in each direction) and further that pseudoalgebras are unbiased monoidal functors. Again the 22-monad satisfies the coherence conditions, which gives a similar commuting square as that one above where the vertical functors are unbiased monoidal equivalences, saying that every unbiased monoidal functor is equivalent to a strict monoidal functor.

Even niftier is that all of the Cat\mathbf{Cat}-operads mentioned above, that give the various flavours of monoidal category, are also 22-cartesian and we can go through the same process with those.

Re: Codescent Objects and Coherence

Street’s 420 paper ‘Fibrations and Yoneda’s lemma in a 2-category’ describes fibrations in a 2-category as the pseudoalgebras of a 2-monad; the strict algebras are the split fibrations. Power’s coherence theorem then gives the result that in Cat every fibration is equivalent to a split fibration. Do the results of this week’s paper apply to fibrations in more general 2-categories? When does the 2-monad preserve codescent objects? The 2-monad acts by composition of spans / by pullback, so this is perhaps some kind of exactness property.

Re: Codescent Objects and Coherence

Re: Codescent Objects and Coherence

A nice example to play around with is the 2-monad whose strict algebras are 2-functors from CC to CatCat.

Ie. given a small 2-category CC one has the set of objects obCobC and inclusion obC→CobC\to C which gives rise to restriction U:[C,Cat]→[obC,Cat]U:[C,Cat]\to [obC,Cat]. Because obCobC is discrete the left adjoint, left Kan extension, has the easy formula FX(c)=ΣjC(j,c)×XjFX(c)=\Sigma_{j}C(j,c) \times Xj.

UU is monadic and the monad T=UFT=UF is cocontinuous and so preserves codescent objects and satisfies the various coherence theorems.

What is nice about this particular 2-monad is that you can easily check that the pseudoalgebras for TT are precisely pseudofunctors, and not something unbiased, so that the coherence result then asserts exactly that each pseudofunctor from CC to CatCat is equivalent to a 2-functor.

Re: Codescent Objects and Coherence

Another interesting thing about that example is that UU is also comonadic, via a continuous comonad. Thus, the dual of the first type of coherence theorem is also true, i.e. the inclusion from pseudofunctors into 2-functors also has a right adjoint.

Re: Codescent Objects and Coherence

Here’s an exercise: does the theory of this paper imply that every symmetric monoidal category is equivalent to a strict symmetric monoidal category, where associators and symmetries are all identities? Lack references an argument of Isbell reported in Mac Lane’s book (on p. 160) which should show the underlying category of the strict replacement can’t be isomorphic to the one we started with in general.

So the question is: does the 2-monad for commutative monoids in Cat\mathbf{Cat} preserve coisoinserters and coequifiers?

If this does work out, what does the strict replacement of a symmetric monoidal category look like? An interesting test case would be the monoidal category of graded vector spaces, which admits at least two interesting symmetries: one has σ(x⊗y)=y⊗x\sigma(x\otimes y) = y \otimes x while the other has σ(x⊗y)=(−1)degxdegyy⊗x\sigma(x\otimes y) = (-1)^{\mathrm{deg} x \, \mathrm{deg} y} y \otimes x. It would be interesting to compare the strictifications for these two different symmetries.

Re: Codescent Objects and Coherence

Re: Codescent Objects and Coherence

Isn’t the category of graded vector spaces (with the “non-trivial” symmetry) a counterexample? Consider σV,V:V⊗V→V⊗V\sigma_{V,V} : V \otimes V \to V \otimes V. If the category in question were equivalent (as a symmetric monoidal category) to a strict symmetric monoidal category, then by naturality, σV,V=id\sigma_{V,V} = \mathrm{id}. But this is not the case for a general VV.

Re: Codescent Objects and Coherence

This should mean that there is no coherence theorem of the second type for the commutative monoid monad TT on Cat\mathbf{Cat}, i.e. not every pseudoalgebra is equivalent to a strict one. It doesn’t rule out a coherence theorem of the first type, i.e. having a left 2-adjoint to the forgetful functor T−Algs→Ps−T−AlgT-\mathrm{Alg}_s \to \mathrm{Ps}-T-\mathrm{Alg}.

But of course, Lack shows that if TT preserves codescent objects then a coherence theorem of the second type holds. So TT must fail to preserve codescent objects; in particular it must fail to preserve either coisoinserters or coequifiers. Which one is it?

The natural next question is: does TT preserve filtered colimits? If it does, then Lack’s Thm 2.4 shows that TT satisfies a coherence theorem of the first type, and as a bonus, we get a simpler counterexample to Lack’s “missing coherence theorem”. It seems to me that TT must preserve filtered colimits because it is the monad for certain algebras with finite arities, right?

Re: Codescent Objects and Coherence

One should not confuse strict symmetric (strict) monoidal categories with symmetric strict monoidal categories. The former are literally internal commutative monoids whereas the latter are the things that symmetric monoidal categories are pseudo-versions of. (To reiterate, a pseudoalgebra for the commutative monoid monad is not a symmetric monoidal category!)

But your intuition is correct: the symmetric strict monoidal category monad does indeed preserve filtered colimits, since it is given by the formula TX=∐n≥0Xn//SnT X = \coprod_{n \ge 0} X^n // S^n, where Xn//SnX^n // S_n is the pseudo-quotient of XnX^n by the canonical SnS_n-action, which is just a certain weighted colimit.

Re: Codescent Objects and Coherence

It’s been a while since I thought about this, but I believe there is a coherence theorem of the second type for the commutative monoid monad on CatCat. But, as Zhen says, even if that’s the case, it doesn’t tell you that symmetric monoidal categories can be strictified to commutative monoids in CatCat, since the pseudoalgebras for the commutative monoid monad are not symmetric monoidal categories. It’s a fun exercise to work out what they are — they’re kind of weird-looking!

Re: Codescent Objects and Coherence

Nice post Alex!

I have a question related to the motivation of lax coherence data. For 1-monads every algebra has a standard presentation as quotient of a free algebra. So is there some way to present a lax algebra for a 2-monad as some kind of colimit of a standard diagram?

Re: Codescent Objects and Coherence

To answer your question: sort of! I imagine what I’m about to say could be done by taking some form of lax limit, rather than a pseudo-limit. As you say, an algebra (A,a)(A,a) for a monad TT can be seen as the colimit of the reflexive coequalizer
⟶μAT2A⟵TηATA.⟶Ta
\begin{array}{ccc}
\quad & \overset{\mu_A}{\longrightarrow} & \quad \\
T^2A & \overset{T\eta_A}{\longleftarrow} & TA. \\
\quad & \overset{Ta}{\longrightarrow} & \quad \\
\end{array}
Now if we have a pseudoalgebra (or lax algebra) then we obviously can’t do this as we know that Ta∘TηA≠1Ta \circ T\eta_A \neq 1, there is a 22-cell in there. This is why we have the coherence data. What we can do with this is look at a certain kind of colimit called a pseudocoequalizer - this is seen in the paper Beck’s theorem for pseudo-monads by Le Creuer, Marmalejo, and Vitale. The relevant definition is given near the start of section 22 in that paper, whilst Lemma 2.32.3 shows that the morphism a:TA→Aa : TA \rightarrow A is the pseudocoequalizer of the pseudoalgebra’s coherence data.

Re: Codescent Objects and Coherence

Unfortunately, calling that kind of colimit a “pseudo-coequalizer” is incorrect, since it is not the pseudo-fication of a coequalizer. A pseudo-coequalizer would be given f,g:A⇉Bf,g:A\;\rightrightarrows\; B and consider the object CC universally equipped with maps p:A→Cp:A\to C and q:B→Cq:B\to C and isomorphisms qf≅pq f \cong p and qg≅pq g \cong p. The correct name is… “codescent object”. (-:

Re: Codescent Objects and Coherence

The “coherence theorem of the first type” that Steve proves, describing a bijection between pseudomorphisms A→BA \to B and strict morphisms A′→BA' \to B can be understood as construction some sort of cofibrant replacement – at least in the case where AA is also a strict algebra.

Can this analogy between “cofibrancy” and “coherence” be pushed any further?

Re: Codescent Objects and Coherence

The categorification of various coherence theorems into the ones ‘of first type’ (correspondence between weak and strict morphisms) and ‘of second type’ (equivalence between strict and weak algebras) is very interesting. In particular, the addition of a coherence result of first type to Power’s coherence result of second type was achieved by proving that the established equivalences between pseudo and strict algebras constitute the unit of the pursued adjunction. Of course, the enhanced factorisation system plays an important role in this proof, but is this phenomenon something we would expect? Are the coherence theorems usually/always connected via such a relation?

Re: Codescent Objects and Coherence

That’s a very interesting question! Actually, I think it’s two questions:

Does there exist a 2-monad TT for which every pseudoalgebra is equivalent to a strict one, but for which the inclusion TAlg→PsTAlgT Alg \to Ps T Alg does not have a left adjoint?

Does there exist a 2-monad TT for which every pseudoalgebra is equivalent to a strict one, and the inclusion TAlg→PsTAlgT Alg \to Ps T Alg has a left adjoint, but the components of the unit are not equivalences?

I suspect the answer to the first question is yes, since without any cocompleteness hypotheses you can cook up all sorts of terrible things like Steve’s example (3.1). But I wouldn’t be surprised if the answer to the second were no.

Re: Codescent Objects and Coherence

With great consideration and apologizing for the occasion , a very valuable standpoint in descriptive universalization is defined in ’ Isomorphic formulae in classical propositional logic ’ by Dòsen and Petric in Mat Log Q 2012 ; 58 ; 1-2 : 5-17 , explaining the syntactic characteristics of pairs of isomorphic formulae in the spirit of coherence results in monoidal category .